Points on curves in small boxes and applications

Abstract

Abstract. We introduce several new methods to obtain upper bounds on the number of solutions of the congruenceswith a prime p and a polynomial f , where (x, y) belongs to an arbitrary square with side length M . We use these results and methods to derive non-trivial upper bounds for the number of hyperelliptic curvesover the finite field F p of p elements, with coefficients in a 2g-dimensional cubethat are isomorphic to a given curve and give an almost sharp lower bound on the number of non-isomorphic hyperelliptic curves with coefficients in that cube. Furthermore, we study the size of the smallest box that contain a partial trajectory of a polynomial dynamical system over F p .

Groups generated by iterations of polynomials over finite fields

“…We also recall that in the case of intervals, the results of [4][5][6][7] apply to much shorter segments of trajectories. It is certainly a natural question to extend the results and methods of [4][5][6][7] to the case of segments of trajectories in subgroups.…”

“…It is certainly a natural question to extend the results and methods of [4][5][6][7] to the case of segments of trajectories in subgroups.…”

Additive Combinatorics over Finite Fields: New Results and Applications

“…Several more results of this type, that apply to large sets and are also based on the method of Garaev [96] have been given by Cilleruelo, Garaev, Ostafe & Shparlinski [68]. The ideas of [68] have found application in the study of the distribution of some families of algebraic curves in isomorphism classes, see [63,69,70].…”

“…This direction has been continued in [60,61,63,68] using further tools and ideas from additive combinatorics.…”

Elliptic Curves over Finite Fields: Number Theoretic and Cryptographic Aspects

“…The method is related to several recent results on the distribution of solutions to polynomial congruences in very small boxes, see [30][31][32] for further references. Both the bound of [32] on I (R, S; M) and some estimates of [30][31][32] on the density of solutions of polynomial congruences have been improved and generalised in [28].…”